net Student Solutions Manual for SINGLE VARIABLE CALCULUS SEVENTH EDITION DANIEL ANDERSON University of Iowa www. COLE Anoka-Ramsey Community College DANIEL DRUCKER Wayne State University Australia. United States www.net © 2012 Brooks/Cole, Cengage Learning ISBN-13: 978-0-8400-4949-0 ISBN-10: 0-8400-4949-8 ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or Brooks/Cole used in any form or by any means graphic, electronic, or 20 Davis Drive mechanical, including but not limited to photocopying, record- Belmont, CA 94002-3098 ing, scanning, digitizing, taping, Web distribution, information USA networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Cengage Learning is a leading provider of customized Copyright Act, without the prior written permission of the learning solutions with office locations around the globe, publisher except as may be permitted by the license terms including Singapore, the United Kingdom, Australia, below.
Mexico, Brazil, and Japan. Locate your local office at: www.com/global For product information and technology assistance, contact us at Cengage Learning products are represented in Cengage Learning Customer & Sales Support, Canada by Nelson Education, Ltd. 1-800-354-9706 For permission to use material from this text or product, For your course and learning solutions, visit submit all requests online at www.com/permissions www.com/brookscole Further permissions questions can be e-mailed to permissionrequest@cengage.com Purchase any of our products at your local college store or at our preferred online store www.net Printed in the United States of America 1 2 3 4 5 6 7 8 15 14 13 12 11 www.net ■ PREFACE This Student Solutions Manual contains strategies for solving and solutions to selected exercises in the text Single Variable Calculus, Seventh Edition, by James Stewart. It contains solutions to the odd-numbered exercises in each section, the review sections, the True-False Quizzes, and the Problem Solving sections, as well as solutions to all the exercises in the Concept Checks.
This manual is a text supplement and should be read along with the text. You should read all exercise solutions in this manual because many concept explanations are given and then used in subsequent solutions. All concepts necessary to solve a particular problem are not reviewed for every exercise. If you are having difficulty with a previously covered concept, refer back to the section where it was covered for more complete help.
A significant number of today’s students are involved in various outside activities, and find it difficult, if not impossible, to attend all class sessions; this manual should help meet the needs of these students. In addition, it is our hope that this manual’s solutions will enhance the understand- ing of all readers of the material and provide insights to solving other exercises. We use some nonstandard notation in order to save space. If you see a symbol that you don’t recognize, refer to the Table of Abbreviations and Symbols on page v.
We appreciate feedback concerning errors, solution correctness or style, and manual style. Any comments may be sent directly to jeff.edu, or in care of the publisher: Brooks/Cole, Cengage Learning, 20 Davis Drive, Belmont CA 94002-3098.net We would like to thank Stephanie Kuhns and Kathi Townes, of TECHarts, for their production services; and Elizabeth Neustaetter of Brooks/Cole, Cengage Learning, for her patience and sup- port. All of these people have provided invaluable help in creating this manual. Cole Anoka-Ramsey Community College James Stewart McMaster University and University of Toronto Daniel Drucker Wayne State University Daniel Anderson University of Iowa iii www.net ■ ABBREVIATIONS AND SYMBOLS CD concave downward CU concave upward D the domain of f FDT First Derivative Test HA horizontal asymptote(s) I interval of convergence I/D Increasing/Decreasing Test IP inflection point(s) R radius of convergence VA vertical asymptote(s) CAS = indicates the use of a computer algebra system.
H = indicates the use of l’Hospital’s Rule. j = indicates the use of Formula j in the Table of Integrals in the back endpapers. s = indicates the use of the substitution {u = sin x, du = cos x dx}.net = c indicates the use of the substitution {u = cos x, du = − sin x dx}.net ■ CONTENTS ■ DIAGNOSTIC TESTS 1 1 ■ FUNCTIONS AND LIMITS 9 1.1 Four Ways to Represent a Function 9 1.2 Mathematical Models: A Catalog of Essential Functions 14 1.3 New Functions from Old Functions 18 1.4 The Tangent and Velocity Problems 24 1.5 The Limit of a Function 26 1.6 Calculating Limits Using the Limit Laws 29 1.7 The Precise Definition of a Limit 34 1.8 Continuity 38 Review 43 www.net Principles of Problem Solving 51 2 ■ DERIVATIVES 53 2.1 Derivatives and Rates of Change 53 2.2 The Derivative as a Function 58 2.4 Derivatives of Trigonometric Functions 71 2.5 The Chain Rule 74 2.7 Rates of Change in the Natural and Social Sciences 85 2.9 Linear Approximations and Differentials 94 Review 97 Problems Plus 105 vii www.net viii ■ CONTENTS 3 ■ APPLICATIONS OF DIFFERENTIATION 111 3.1 Maximum and Minimum Values 111 3.2 The Mean Value Theorem 116 3.3 How Derivatives Affect the Shape of a Graph 118 3.4 Limits at Infinity; Horizontal Asymptotes 128 3.5 Summary of Curve Sketching 135 3.6 Graphing with Calculus and Calculators 144 3.9 Antiderivatives 167 Review 172 Problems Plus 183 4 ■ INTEGRALS 189 4.1 Areas and Distances 189 www.3 The Definite Integral 194 The Fundamental Theorem of Calculus 199 4.4 Indefinite Integrals and the Net Change Theorem 205 4.5 The Substitution Rule 208 Review 212 Problems Plus 217 5 ■ APPLICATIONS OF INTEGRATION 219 5.1 Areas Between Curves 219 5.3 Volumes by Cylindrical Shells 234 5.5 Average Value of a Function 241 Review 242 Problems Plus 247 www.net CONTENTS ■ ix 6 ■ INVERSE FUNCTIONS: Exponential, Logarithmic, and Inverse Trigonometric Functions 251 6.2 Exponential Functions and 6.2* The Natural Logarithmic Their Derivatives 254 Function 270 6.3* The Natural Exponential Functions 261 Function 277 6.4 Derivatives of Logarithmic 6.4* General Logarithmic and Functions 264 Exponential Functions 283 6.5 Exponential Growth and Decay 286 6.6 Inverse Trigonometric Functions 288 6.8 Indeterminate Forms and L’Hospital’s Rule 298 Review 306 Problems Plus 315 7 www.net ■ 319 TECHNIQUES OF INTEGRATION 7.1 Integration by Parts 319 7.4 Integration of Rational Functions by Partial Fractions 334 7.5 Strategy for Integration 343 7.6 Integration Using Tables and Computer Algebra Systems 349 7.8 Improper Integrals 361 Review 368 Problems Plus 375 8 ■ FURTHER APPLICATIONS OF INTEGRATION 379 8.2 Area of a Surface of Revolution 382 8.3 Applications to Physics and Engineering 386 www.4 Applications to Economics and Biology 393 8.5 Probability 394 Review 397 Problems Plus 401 9 ■ DIFFERENTIAL EQUATIONS 405 9.1 Modeling with Differential Equations 405 9.2 Direction Fields and Euler’s Method 406 9.4 Models for Population Growth 417 9.6 Predator-Prey Systems 425 Review 427 Problems Plus 433 10 www.net ■ PARAMETRIC EQUATIONS AND POLAR COORDINATES 437 10.1 Curves Defined by Parametric Equations 437 10.2 Calculus with Parametric Curves 443 10.4 Areas and Lengths in Polar Coordinates 456 10.6 Conic Sections in Polar Coordinates 468 Review 471 Problems Plus 479 11 ■ INFINITE SEQUENCES AND SERIES 481 11.3 The Integral Test and Estimates of Sums 495 11.4 The Comparison Tests 498 www.net CONTENTS ■ xi 11.6 Absolute Convergence and the Ratio and Root Tests 504 11.7 Strategy for Testing Series 508 11.9 Representations of Functions as Power Series 514 11.10 Taylor and Maclaurin Series 519 11.11 Applications of Taylor Polynomials 526 Review 533 Problems Plus 541 ■ APPENDIXES 547 A Numbers, Inequalities, and Absolute Values 547 B Coordinate Geometry and Lines 549 C Graphs of Second-Degree Equations 552 D Trigonometry 554 www.net E G Sigma Notation 558 Graphing Calculators and Computers 561 H Complex Numbers 564 www.net DIAGNOSTIC TESTS Test A Algebra 1. (a) Note that 200 = 100 · 2 = 10 2 and 32 = 16 · 2 = 4 2.net Or: Use the formula for the difference of two squares to see that (d) (2 + 3)2 = (2 + 3)(2 + 3) = 42 + 6 + 6 + 9 = 42 + 12 + 9.
Note: A quicker way to expand this binomial is to use the formula ( + )2 = 2 + 2 + 2 with = 2 and = 3: (2 + 3)2 = (2)2 + 2(2)(3) + 32 = 42 + 12 + 9 (e) See Reference Page 1 for the binomial formula ( + )3 = 3 + 32 + 32 + 3. Using it, we get ( + 2)3 = 3 + 32 (2) + 3(22 ) + 23 = 3 + 62 + 12 + 8. [See Reference Page 1 in the textbook.] (e) The smallest exponent on is − 12 , so we will factor out −12 .net 2 ¤ DIAGNOSTIC TESTS 2 + 3 + 2 ( + 1)( + 2) +2 5. In interval notation, the answer is [−4 3).
Now, ( + 2)( − 4) will change sign at the critical values = −2 and = 4. Thus the possible intervals of solution are (−∞ −2), (−2 4), and (4 ∞). By choosing a single test value from each interval, we see that (−2 4) is the only interval that satisfies the inequality.net TEST B ANALYTIC GEOMETRY ¤ 3 (c) The inequality ( − 1)( + 2) 0 has critical values of −2 0 and 1. The corresponding possible intervals of solution are (−∞ −2), (−2 0), (0 1) and (1 ∞).
By choosing a single test value from each interval, we see that both intervals (−2 0) and (1 ∞) satisfy the inequality. Thus, the solution is the union of these two intervals: (−2 0) ∪ (1 ∞). In interval notation, the answer is (1 7). +1 +1 +1 +1 +1 +1 −4 Now, the expression may change signs at the critical values = −1 and = 4, so the possible intervals of solution +1 are (−∞ −1), (−1 4], and [4 ∞).
By choosing a single test value from each interval, we see that (−1 4] is the only interval that satisfies the inequality. In order for the statement to be true, it must hold for all real numbers, so, to show that the statement is false, pick = 1 and = 2 and observe that (1 + 2)2 6= 12 + 22. (b) True as long as and are nonnegative real numbers. To see this, think in terms of the laws of exponents: √ √ √ = ()12 = 12 12 = .
To see this, let = 1 and = 2, then 12 + 22 6= 1 + 2. To see this, let = 1 and = 2, then 6= 1 + 1. To see this, let = 2 and = 3, then 1 2−3 1 1 6= −. 2 3 1 1 (f ) True since · = , as long as 6= 0 and − 6= 0.
− − Test B Analytic Geometry 1. (b) A line parallel to the -axis must be horizontal and thus have a slope of 0. Since the line passes through the point (2 −5), the -coordinate of every point on the line is −5, so the equation is = −5. (c) A line parallel to the -axis is vertical with undefined slope.
So the -coordinate of every point on the line is 2 and so the equation is = 2. Thus the slope of the given line is = 12. Hence, the slope of the line we’re looking for is also 12 (since the line we’re looking for is required to be parallel to the given line). So the equation of the line is − (−5) = 12 ( − 2) ⇒ + 5 = 12 − 1 ⇒ = 12 − 6.
